Numerical prediction of process-dependent properties of high-performance Ti6Al4 in LS-DYNA

In modern manufacturing processes such as hot forming or additive manufacturing, the workpiece material undergoes very complex thermomechanical load cycles. The local mechanical properties in the component are process-dependent and the result of the different micro-structure evolution mechanisms in the material. Numerical process simulation tools aim to include more and more of these mechanisms in order to improve the accuracy of the simulations. The mechanical strength of high-performance materials such as Ti-6Al-4V depends on microstructural parameters, which are influenced by the temperature and strain histories. This contribution puts forward an implementation of a new generalized internal variables material model *MAT_GENERALIZED_PHASECHANGE in LS-DYNA. The evolution of internal variables such as phase fractions, grain size and dislocation densities can be predicted by evolution equations, and combined with yield stress models taking the contribution of the phases, grain sizes (Hall-Petch effect), and the dislocation density into account to predict the resulting mechanical properties of the processed material. The benefits of the implementation in the commercial software LS-DYNA is the possibility to solve complex coupled problems. For example, the new material law can be used to simulate hybrid manufacturing processes like forging and an additional additive manufacturing process, where changes in microstructure are highly coupled and important for the part properties.


Intr Introduction oduction
High-performance materials obtain their characteristic properties such as strength only by adapted production processes that control the microstructure in the material. Consequently, there is a need for numerical process simulations of production processes to predict the mechanical properties resulting from the manufacturing process.
In the automotive industry, the distributions of pre-strain, thinning and pre-damage as predicted by a numerical forming simu-lation are often taken as initial values for the crashworthiness analysis. Modern commercial simulation packages, such as LS-DYNA, are, thus, essential throughout the automotive development process. Their capabilities are not limited to this field.
For other industries, the numerical process simulation gains importance, too. Due to the small material yield in conventional processing of titanium alloys, e.g. Ti-6Al-4V, several innovative processes have been proposed recently.
While the conventional processes lead to known material properties, this is not the case for processes such as additive manufacturing followed by hot isostatic pressing (HIP) [1] and combinations of AM and metal forming [2,3]. The material properties of Ti-6Al-4V depend on the microstructure generated by phase transformations and evolution of internal variables such as dislocation densities and grain size. In most commercial tools, tailored formulations for the phase transformation kinetics are only available for a limited number of processes and materials, like the hot stamping of the manganese-boron-steel 22MnB5 [4]. To overcome this limitation, the material model *MAT_254 / *MAT_GENERALIZED_PHASE_CHANGE provides a more general framework in terms of materials and processes in the commercial multi-physics FE-Solver LS-DYNA. It has been successfully been applied to, first, hot stamping and welding applications with steel alloys [5] and, second, to aluminum alloys [6].
Recent enhancements aim for process simulations with Ti-6Al-4V [7]. This material model allows working with up to 24 different phases, where it is up to the user to define the physical representation of each phase and the phase transformations to be considered. The resulting properties are obtained from the phase fractions and properties of each phase using a mixture rule.
Industrial process simulation is performed on the macroscopic (process) scale and therefor the morphology inside the material cannot be considered directly. Instead, so called internal variables are used to quantify the effects. This contribution shows a gen-eralized approach to include internal variables into macroscopic process simulations with LS-DYNA. Focus is put on necessary variables for Ti-6Al-4V, i.e., the evolution of phases, grain size and dislocation density. Other aspects such as precipitations and their morphology, can also be addressed by this approach, but is not of importance for Ti-6Al-4V.
The next section gives a short overview on *MAT_254, followed by a summary of the model of Bambach et al. [8] and its implementation into the material law. The application of this extended material on different test cases shows the functionality of the implementation in the third section. The commercial Finite Element Code LS-DYNA is used in a wide range of engineering problems. Due to its "one-codestrategy" it enables the users to easily simulate coupled problems like the thermo-mechanical forming processes. This includes phase trans-formations, which occur in materials such as steel and titanium alloys, with the LS-DYNA material model *MAT_254 / *MAT_GENERALIZED_PHASE-CHANGE. Fig. 1 shows the LS-Dyna input structure for this material.
Numerical prediction of process-dependent properties of high-performance Ti6Al4 in LS-DYNA 1496/2 where P is an arbitrary property and denotes the concentration of phase .
The second card controls the annealing algorithm, accounts for thermal expansion, and defines a form of internal sub-cycling to resolve non-linear effects. As none of these features is used here, the respective parameters are not discussed.
The third and fourth cards specify the phase transformation models. Parameter influences the shape of the data referred to by the parameters in card 3 and 4 as it determines the input structure to be a by -matrix, which is read by the material model as "from phase … to phase …" as is shown exemplarily in Fig. 2   The parameter PTLAW on the third card refers to such a matrix. It contains information between which of the phases transformations can occur during cooling, heating or both and determines the employed transformation model.
Of particular interest are the transformation laws associated with the values of "5", "6", "7", "8", and "9", which are tailored for Ti-6Al-4V. The implementation follows the modelling approach presented by Charles-Murgau et al. [9,10] and Klusemann and Bambach [11] and has been validated in a previous contribution [5]. As sketched in Fig. 2 (b), four phases are defined (N=4) in the case of Ti-6Al-4V. They can be identified as -phase, martensitic/massive , Widmannstätten , and grain boundary , respectively. Recovery of the -phase and Widmannstätten from martensitic and massive in heating conditions are described by transformation models "5" and "6". The successive parabolic dissolution from the -phases into -phase during heating is controlled by the transformation models "7" and "8". When the material cools down, the -phase is transformed into the different -phases. This is modelled by an extended Koistinen-Marburger approach ("9") and a JMAK approach ("2"), respectively.
2.2 Yield str 2.2 Yield stress computation in hotf ess computation in hotforming conditions f orming conditions for Ti-6Al-4V or Ti-6Al-4V By default, the yield stress is calculated by a linear mixture rule as in (1). Thus, the overall yield stress depends on the current phase fractions and the current values of the yield stresses of the individual phases: The values of phase contributions are functions of plastic strains, strain rates and temperature. Particularly, this last assumption is not justified for Ti-6Al-4V in hot forming conditions according to [8] and [12]: the yield stress Modelling compound materials knows two extrema, which can be interpreted either as a series connection (uniform stress model) or a parallel connection of springs (uniform strain model), representing the compounds. As the uniform strain model essentially leads to a linear mixture rule for the yield stresses as in (1), only this model is discussed in this contribution: with and being the yield stresses of the -and -phases and their fractions and . With the phase mapping shown in Fig. 2 (b) comparison of (2) and (3) allows for the following identifications: Neither nor as defined in [8] fit into the standard description for the yield stress of a phase as provided by *MAT_254 and, consequently, an enhancement of the material model is needed to simulate hot forming applications with titanium. The contri-bution is derived from experiments above the -transus temperature, where only -phase is found. It results in a phenomeno-logical model, cf. [13], that reads as follows: Both stress contributions 1 and 2 , are functions of an equivalent strain quantity and the Zener-Hollomon parameter , which introduces a dependence on the strain rate ̇ and the temperature by its definition: For the complete and rather lengthy expressions of 1 and 2 the reader is referred to the literature [8]. This flexible approach can also be used to define the yield stress for the -phases. However, the definition adopted from [14], requires additional enhancements. The yield stress is assumed to be a combination of a thermal stress contribution * and an athermal stress contribution : with being the constant Taylor factor. The thermal part * is a function of strain ratėand the temperature and can, thus, be defined with the new reference to an analytical expression as discussed above. For the algebraic expression of * , the user is referred to [8]. The athermal stress requires further discussion as it is the result of different hardening mechanisms: where 0 is the mechanical threshold stress, 2 a material constant, the shear modulus, the magnitude of the Burgers Vector, the dislocation density, the Hall-Petch coefficient and the average lath thickness of -platelets.
In Eq. (9), the second term can be interpreted as work-hardening effect and the last term as grain size effect. These

Int Internal v ernal variable definition in *MA ariable definition in *MAT_254 T_254
The material input card in Fig. 1 represents the status of the material implementation. Parameters associated with the recent changes are highlighted. Parameter NIVAR defines the number of internal variables, but only up to four internal variables can currently be considered. For each internal variable, two additional lines of input are required.
The first additional entry IVAROPT defines the evolution equations to be used for the calculation. Five options for IVAROPT are available: • "1" -use a *DEFINE_FUNCTION (a user defined function) • "2" -use a grain size calculation for 22MnB5 (like in *MAT_244/248 in LS-DYNA) • "3" -calculates the percentage of globular -phase in Ti-6Al-4V, as in [8] • "4" -calculates the dislocation density in Ti-6Al-4V, as in [8] • "5" -calculates the Hall-Petch coefficient in Ti-6Al-4V, as in [8] Option "1" offers the most flexibility and option "2" is tailored for UHS steels. In the following paragraphs, options "3", "4" and "5" will be discusses in some detail as these are connected to the hot-forming of Ti-6Al-4V. The meaning of the evolution parameters IVARPAR1 to IVARPAR14 in the input depends on the choice of IVAROPT. In the following, the notation is used in the equations for the -th parameter IVARPARi to shorten the expressions. Although the flow stress in (6) only contains the internal variables and , there is an implicit dependence on a third internal variable, namely the volume fraction of globularized -phase . Its evolution interacts with the evolution of the dislocation density.
Computation is invoked with IVAROPT=3 and starts with determining the critical strain needed for the start of the globularization: where the parameter 4 can be interpreted as ratio of an activation energy and the universal gas constant : 4= / . If this critical value is exceeded, globularization follows an ordinary differential equation (ODE) with initial value 0 that is input as parameter INITVAL. The ODE depends on strain , strain rate, material constants 0, 1, 2, the average -plate thickness , the grain boundary mobility , and the activation force per unit area , resulting in It is important to note that mobility and activation force are not constants but functions of temperature , shear modulus and even of the dislocation density . The actual expressions for and are omitted here and can be found in [8]. Introducing the expressions into (11) and merging the material constants, the following form of the evolution equation can be obtained, which is used in the material model: The dislocation density is not an intrinsic parameter of the material model, but it needs to be defined as internal variable as described in the following subsection. Therefore, additional information is needed for evaluation of (12).
Parameter IVARPAR12 thus corresponds to the identification number of that internal variable that represents the dislocation density.

Dislocation density (IV Dislocation density (IVAR AROPT=4) OPT=4)
The work hardening part in (9) mainly dependents on the dislocation density . The initial value 0 is defined by the second entry INITVAL of the first additional line. For the evolution equation that is activated by IVAROPT=4, a critical strain again plays a role. It is calculated using (10)  The previous equation shows that the evolution explicitly depends on globularization fraction . The parameter IVARPAR12 determines the identification number of the internal variable "globularization fraction". Finally, the parameter 2 in (13) depends on temperature and strain rate: The parameter 9 can be interpreted as ratio of an activation energy and the universal gas constant : 9= / .
Furthermore, it is worth mentioning that in [8] the same activation energy value has been used for the critical strain in (10) and for the parameter 2 in (19), equivalent to choosing 4= 9.

Hall-P Hall-Pet etch coefficient (IV ch coefficient (IVAR AROPT=5) OPT=5)
The final term in (9)  First, only the -phase is considered because this allows to validate the new interface to user-defined flow-stress input, by defining SIGY1 to refer to a function representing in (6). Fig. 3 shows, that the new yield stress interface agrees precisely with the reference, proofing the validity of the implementation.
Each -phase uses the same yield stress definition. Thus, parameters SIGY2, SIGY3, and SIGY4 refer to the same function id representing in (8). As required by the expression, three internal variables are defined (NIVAR=3), for globularization, disloca-tion density and Hall-Petch coefficient, respectively. The flow curves predicted by the model are shown in Fig. 4. They show a perfect qualitative agreement with the reference and a very good quantitative fit.
Parameters, which are not stated in [8], were estimated for the presented results and explain the minor differences. Finally, the features of *MAT_254 for phase evolution kinetics and flow stress prediction are combined. Assuming an initial mixture of (10%), (88.5%) and (1.5%), elements are heated within 120s to 850°C, 900°C and 950°C, respectively. Fig. 5 (a) exemplarily shows the phase evolution during heating to 950°C. Flow stress results for combined -and -phase for the different temperatures at a strain rate of 1/s are given in Fig. 5 (b). Again, the results are in good agreement to the literature reference [8].  The validity of this approach for complex processes such as hybrid forming and welding applications like [17] will be the subject of further investigations.